20 research outputs found
Mean-field backward stochastic differential equations: A limit approach
Mathematical mean-field approaches play an important role in different fields
of Physics and Chemistry, but have found in recent works also their application
in Economics, Finance and Game Theory. The objective of our paper is to
investigate a special mean-field problem in a purely stochastic approach: for
the solution of a mean-field backward stochastic differential equation
driven by a forward stochastic differential of McKean--Vlasov type with
solution we study a special approximation by the solution
of some decoupled forward--backward equation which coefficients are governed by
independent copies of . We show that the convergence speed
of this approximation is of order . Moreover, our special choice of
the approximation allows to characterize the limit behavior of
. We prove that this triplet converges in law to
the solution of some forward--backward stochastic differential equation of
mean-field type, which is not only governed by a Brownian motion but also by an
independent Gaussian field.Comment: Published in at http://dx.doi.org/10.1214/08-AOP442 the Annals of
Probability (http://www.imstat.org/aop/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Mean-field backward stochastic differential equations and related partial differential equations
In [R. Buckdahn, B. Djehiche, J. Li, S. Peng, Mean-field backward stochastic differential equations. A limit approach. Ann. Probab. (2007) (in press). Available online: http://www.imstat.org/aop/future_papers.htm] the authors obtained mean-field Backward Stochastic Differential Equations (BSDE) associated with a mean-field Stochastic Differential Equation (SDE) in a natural way as a limit of a high dimensional system of forward and backward SDEs, corresponding to a large number of "particles" (or "agents"). The objective of the present paper is to deepen the investigation of such mean-field BSDEs by studying them in a more general framework, with general coefficient, and to discuss comparison results for them. In a second step we are interested in Partial Differential Equations (PDE) whose solutions can be stochastically interpreted in terms of mean-field BSDEs. For this we study a mean-field BSDE in a Markovian framework, associated with a McKean-Vlasov forward equation. By combining classical BSDE methods, in particular that of "backward semigroups" introduced by Peng [S. Peng, J. Yan, S. Peng, S. Fang, L. Wu (Eds.), in: BSDE and Stochastic Optimizations; Topics in Stochastic Analysis, Science Press, Beijing (1997) (Chapter 2) (in Chinese)], with specific arguments for mean-field BSDEs, we prove that this mean-field BSDE gives the viscosity solution of a nonlocal PDE. The uniqueness of this viscosity solution is obtained for the space of continuous functions with polynomial growth. With the help of an example it is shown that for the nonlocal PDEs associated with mean-field BSDEs one cannot expect to have uniqueness in a larger space of continuous functions.Mean-field models McKean-Vlasov equation Backward stochastic differential equations Comparison theorem Dynamic programming principle Viscosity solution